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Questions tagged [riemann-integration]

The Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. For many functions and practical applications, the Riemann integral can be evaluated by the fundamental theorem of calculus or approximated by numerical integration.

Loosely speaking, the Riemann integral is the limit of the Riemann sums of a function as the partitions get finer. If the limit exists then the function is said to be Riemann-integrable. The Riemann sum can be made as close as desired to the Riemann integral by making the partition fine enough.

The Riemann integral is unsuitable for many theoretical purposes. Some of the technical deficiencies in Riemann integration can be remedied with the Riemann–Stieltjes integral, and most disappear with the Lebesgue integral.

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Equivalence statement of Riemann integrable conditions.

The following two is equivalent. Let $f$ be bounded function from $[a,b]$ to $\Bbb R$ (1) $f$ is Riemann integrable and say $\int_{a}^{b}f=A$ (2) for any $\epsilon \gt 0, \exists \delta \gt 0 $. If partion ||P|| $\lt \delta$, P $\in \Bbb…
fivestar
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Let $g: [0,1] \rightarrow \mathbb R$ and $g(x)=\frac{1}{q} $ if $x=\frac{p}{q}$, otherwise $0$.Prove that g is Riemann integrable.

Let $g: [0,1] \rightarrow \mathbb R$ and $g(x)=\frac{1}{q}$ if $x=\frac{p}{q} $ otherwise $0$.Prove that g is Riemann integrable. Hint: Analyze the discontinuities of $g$. Prove that if $x$ is irrational and if $\frac{p_n}{q_n}$ is a sequence of…
User124356
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Suppose $f \geq 0$, if continuous on [a,b] and $\int_{a}^{b} f(x)dx=0$. Prove that $f(x)=0$ for all $x\in [a,b]$.

My attempt: Let P the partition of $[a,b]$. Let $x_{i}^{*} \in [x_{i-1},x_{i}]$ and f is non negative in $[x_{i-1},x_{i}]$. Since f is continuous on $[a,b]$, then $f$ is R-integrable with $\int_{a}^{b} f(x)dx=0$. $$\displaystyle \lim_{n\rightarrow…
User124356
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To prove that the integral of the function lies between the lower darboux sum and the upper darboux sum

m = inf(f(x): x is a member of [a,b]), M = sup(f(x): x is a member of [a,b]) suppose that the function is Riemann integrable Prove the following $$ m(b-a)\leq {\int_{a}^{b}} f(x) \, \mathrm{d}x \leq M(b-a) $$ Therefore what i have done was take…
John
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Riemann Integrable Question

Hi I am confused about the following question. I am trying to understand the conditions for when the rationals intersect the closed interval. I know that $\mathbb{Q}$ is a subset of $\mathbb{R}$ but don't fully grasp how the function is set up. Let…
John
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Riemann-integral for a function that has infinitely many discontinuity points

The question is following: Let $f:[0,1]\rightarrow \mathbb{R}.$ $f(x)=x,$ if $x=1/n, n\in\mathbb{N}$ $f(x)=0,$ otherwise. Is $f$ Riemann-integrable? If it is, what is its value? I know that the basic idea of Riemann-integral is to find two…
jte
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Riemann-Stieltjes Integral - Changing Order of Integration

I am new to Riemann-Stieltjes integral. I want to ask a very basic question regarding changing the order of integration. Let $ t > 0 $ and I have an integral that looks like this $$ \int_\mathbb{R} \int_0^t f(g(x)) dx dg(x). $$ What is the condition…
Ben
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Relaxing hypotheses for the mean value theorem for integrals

If a function $f$ is continuous in $[0,\Delta]$ it is pretty easy to prove that $$ \exists c\in(0,\Delta):\frac{1}{\Delta}\,\int_{0}^{\Delta}f(t)dt=f(c) $$ It is enough to apply Lagrange's to the function $F(t)=\int_0^tf(s)ds$. Is it possible to…
AlmostSureUser
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Sufficient condition for Riemann integrability

There are many ways to define the Riemann Integral. I am using this one, where I denote $\sigma(f,P^{*})$ the Riemann Sum relative to a tagged partition $P^{*}$: $\textbf{Definition}$ We say that a function $f:[a,b] \to \mathbb{R}$ is…
joelittledrop
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Question about Riemann integral property proof

I just read the following property and proof: Let $I = [a_1, b_1] \times ... \times [a_n, b_n]\subset \mathbb{R}^n$ and $f,g : I \rightarrow \mathbb{R}$ integrable functions such that $f \leq g$. Then $\int_I f \leq \int_I g$. Proof: It is a…
Yagger
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Proving a function is Riemann integrable using Riemann criterion

Let $I=[0,1]\times[0,1]$ and $f:I \rightarrow \mathbb{R}$, $f(x) =\begin{cases} 1&\text{if } (x,y)=(\frac{1}{2},\frac{1}{2})\\ 0&\text{if }(x,y) \neq (\frac{1}{2},\frac{1}{2})\end{cases}$ Prove that f is Riemann integrable in $I$ and its integral…
Yagger
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Is it always possible to choose $x \in (a, b)$ s.t. $\int_a^{x}f=\int_x^{b}f$

I am working out a homework problem about Riemann Integrals and the question is as follows: Suppose that $f$ is integrable on $[a, b]$, then $\exists \ x \in [a, b] s.t. \int_a^{x}f=\int_x^{b}f$. Is it always possible to choose $x$ to be in…
Dylan Zammit
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Function of which the upper sum does not equal the lower sum

For a bounded function $f:[a,b] \rightarrow \mathbb{R}$, we define the lower Riemann-integral of $f$ over $[a,b]$ as $\underline{S}(f)= \sup \{ \underline{S}(f,P) | P$ is a partition of $[a,b] \}$. The upper Riemann-integral is defined as…
simp
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show that this function is not Riemann integrable on $[0,1].$

Let $$f(x) = \begin{cases}x^2 & x\in \Bbb Q \\ x^3 & x \notin \Bbb Q \end{cases}$$ I want to show that this function is not Riemann integrable on $[0,1].$ I though of find the lower and upper $R$-integral and show them not equal but but with…
blabla
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What do we call the $\alpha$ in Riemann Integral?

I am learning the concept of Riemann Integral. $F(x) \text{ sometimes denoted as} \int_a^xfd\alpha$. What I know is $\alpha$ is a function which maps $x\mapsto\Bbb R$(Is this domain and codomain correct?) But I want to know with which term we call…
snapper
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